Nonrobustness Property of the Individual Ergodic Theorem

Nonrobustness Property of the Individual Ergodic Theorem Main laws of probability theory, when applied to individual sequences, have a “robustness” property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length n grows as o(n). The law of iterated logarithm holds if the randomness deficiency grows as o(log log n). We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon–McMillan–Breiman theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Nonrobustness Property of the Individual Ergodic Theorem

Problems of Information Transmission, Volume 37 (2) – Oct 7, 2004
12 pages

/lp/springer_journal/nonrobustness-property-of-the-individual-ergodic-theorem-b8jOf02JQl
Publisher
Springer Journals
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1023/A:1010418008049
Publisher site
See Article on Publisher Site

Abstract

Main laws of probability theory, when applied to individual sequences, have a “robustness” property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length n grows as o(n). The law of iterated logarithm holds if the randomness deficiency grows as o(log log n). We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon–McMillan–Breiman theorem.

Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 7, 2004

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